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A unified formulation to assess multilayered theories for piezoelectric plates

D. Ballhause, M. d’Ottavio, B. Kröplin, E. Carrera

To cite this version:

D. Ballhause, M. d’Ottavio, B. Kröplin, E. Carrera. A unified formulation to assess multilayered theories for piezoelectric plates. Computers and Structures, Elsevier, 2005, 83 (15-16), pp.1217-1235.

�hal-01367092�

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A unified formulation to assess multilayered theories for piezoelectric plates

D. Ballhause, M. DÕOttavio , B. Kro¨plin, E. Carrera

1. Introduction

Because of their self-monitoring and self-adaptive capability, so-called advanced intelligent structures have attracted considerable research over the past few years.

These structures have some distinct advantages over conventional actively controlled structures. Since intelli- gent structures are characterized by distributed actua- tion and sensing systems, more accurate response monitoring and control are possible. Some of the most significant work has concentrated on the development and implementation of actuators and sensors made of piezoelectric materials. Since Curie brothers [1], it is known that there are two basic phenomenaÔdirect and converse effectsÕ, characteristic of piezoelectric materials,

*Corresponding author. Tel.: +39 11 5646836; fax: +39 11 5646899.

E-mail addresses: ballhause@isd.uni-stuttgart.de (D. Ball- hause),erasmo.carrera@polito.it(E. Carrera).

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which permit them to be used as sensors and actuators.

Although piezoelectricity has a long history, its use in actuation and control of light flexible structures for aerospace applications is relatively new. Overview pa- pers are those by Rao and Sunar[2], Chopra[3], Tani et al.[4], and Sunar and Rao[5]. In most of these appli- cations, the piezoelectric materials are used as sensor/

actuator layers by embedding them in a multilayered structure which is often made by advanced anisotropic composite materials.

As pointed out by the already mentioned overview papers, the most important problems arising in the de- sign of intelligent structures embedding piezo-layers are manufacturing, electro-mechanical modeling, opti- mization and control. Among these, the attention is herein focused on plate theories for the electro-mechan- ical modeling of flat, intelligent structures. The accurate description of mechanical and electrical fields in the lay- ers is, in fact, essential in order to both perform a real- istic simulation of direct/converse effect and prevent failure mechanisms of the structure.

The main issue of multilayered piezoelectric construc- tions is related to the possibility of exhibiting different mechanical–electrical properties in the thickness direc- tion. In addition, anisotropic multilayered composites of- ten exhibit both higher transverse shear and transverse normal flexibilities, with respect to in-plane deformabi- lity, than traditional isotropic one-layered ones. As a con- sequence, plate theories which are based on the extension of so-called Kirchhoff (or Classical Lamination Theory, CLT) and Reissner–Mindlin (or First order Shear Defor- mation Theory, FSDT) hypotheses can be ineffective to trace static and dynamic response of piezoelectric plates.

Higher transverse deformabilty demands the inclusion of transverse shear and normal stresses which are dis- carded in classical analyses. Furthermore, transverse dis- continuous mechanical properties cause displacement fieldsu= (ux,uy,uz) which can exhibit a rapid change of their slopes in the thickness direction in correspondence to each layer interface. This is known as the Zig-Zag (ZZ) effect. Nevertheless, equilibrium reasons require Interlaminar Continuity(IC) for transverse stressesrn= (rxz,ryz,rzz). A discussion on theories addressing ZZ and IC has been recently provided by Carrera[6]. As far as electrical variables are concerned, it should be noticed that in order to include a correct description of electrical stiffnesses, the electric field should have at least a linear distribution in thickness direction of the piezo-layers.

As a consequence, at least a parabolic assumption for the electric potential into the layers is required.

Many refined plate theories have been proposed and extended to the electro-mechanical fields. A few exam- ples are mentioned in the following text. Applications of CLT and FSDT to piezoelectric plates were given by Tiersten,[7]and Mindlin[8]. As example of a refined theory the work by Yang and Yu[9]is mentioned. The

electric fields generated by stresses, e.g. electrical stiff- ness, were not considered in these three works. The electro-mechanical coupling was indeed retained in fol- lowing articles. Mitchell and Reddy [10] introduced a layer-wise (LW) description for the electric potential, while an equivalent single layer (ESL) description was retained for displacements. According to Reddy[11], it is intended that the number of displacement variables is kept independent of the number of constitutive layers in the ESL models, while the same variables are indepen- dent in each layer for LW cases. Refined ESL models have been discussed by Benjeddou [12]. ESL formula- tion taking into account ZZ and IC have been discussed by Touratier and Ossadzow-David [13]. A more com- plete discussion of the several contributions on electro- mechanical models of multilayered plates embedding piezo-layers, has been covered by recent exhaustive state-of-the-art articles. Interested readers are addressed to the review papers by Saravanos and Heyliger[14]and by Benjeddou[15].

However, most of the articles proposing refined the- ories for piezoelectric plates restricted the numerical comparison to classical plate theories, such as CLT and FSDT, and to available 3D solutions. Only in some cases other refined theories were included in the verifica- tion of the new method. Such a restriction does not per- mit to give a complete overview and assessment of available theories. The present work aims to contribute to this matter by enlarging the number of the theories available for a single problem: about thirteen theories are, in fact, implemented and compared for different test cases. The most accurate ones coincide with a layer-wise theory based on a fourth order expansion in each layer (LD4). Such a theory leads to a quasi-3D description of mechanical and electrical fields in the layers. The less accurate one coincides with CLT. Eleven further formu- lations with accuracies lying between LD4 and CLT are proposed and compared. These theories are able to cov- er many of the aspects characterizing 2D axiomatic the- ories for multilayered plates, such as ZZ, transverse shear deformation, transverse shear strains effects and to compare LW and ESL description.

In order to meet the mentioned proposals this paper reconsiders recent findings by Carrera[16–19]. In these papers aÔunified formulationÕwas proposed, basing on applications of the Principle of Virtual Displacement (PVD) and ReissnerÕ Mixed Variational Theorem (RMVT) to mechanical problems. Thisunified formula- tionpermits to derive governing equations in terms of a few fundamental nuclei whose expressions do not change by varying the assumptions made for the dis- placement variables in the layer thickness. In this work, these methods are extended to electro-mechanical static and dynamic analyses of multilayered plates embedding piezo-layers. Developments have been restricted to PVD applications. Layer-wise and equivalent single layer

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models with linear up to fourth-order expansion in the plate thickness (z-direction) have been implemented.

Zig-zag effects are described by means of the Murakami zig-zag function [20]. Transverse normal strain effects have also been considered. As far as the electrical poten- tial is concerned, attention has been restricted to a layer-wise description. Results are given for the case of orthotropic layers and simply supported plates loaded by harmonic distributions of mechanical and electrical loadings. Numerical results are given for both free vibra- tion and static response. Verification is made with re- spect to available 3D solutions and an assessment of various plates theories is finally given.

2. Preliminary

A multilayered flat plate consisting of NL layers, which can be piezoelectric or purely elastic is examined.

The laminae are considered homogeneous and perfectly bonded with each other. The notation and the Cartesian coordinate systemx,y,zreferred to the middle surface can be seen inFig. 1. The plate has the lengtha, width band heighth. The indexkis used for the layer number.

The local thickness coordinate of the layer iszk, the non- dimensional correspondent is fk= (2zk/hk). The linear elastic range of the materials and the physical limits of the piezoelectric layer, like Curie temperature and depo- larization potential, are not exceeded by any deforma- tions or loadings. The stresses and strains are represented in engineering notation with the indices 11, 22, 33, 13, 23, and 12. The stiffness coefficientsCeijrefer to these withi,j= 1. . .6. The directions of the piezoelec- tric material are named in the standard manner with 1–

3, axis 3 being the polarization direction of the material.

The polarization axis is assumed parallel to the thickness directionzof the plate. To establish a uniform model- ing, all layers in the laminate are assumed to be piezo- electric, where for purely elastic layers the piezoelectric coefficients are set to zero.

2.1. Constitutive equations

The coupling between the stresses and the electric field in thek-layer is sustained by the direct and converse piezoelectric effect and is represented, according to the IEEE standard[21]with the following constitutive equa- tions in array form:

rk¼CekkekTEk ð1Þ

Dek¼ekkþekEk ð2Þ Boldfaced letters are used for arrays, superscript T de- notes transposition. The vectors r and contain the six stress and strain components, respectively. Dek is the dielectric displacement and Ek the electric field strength with three direction components each

e

Dk¼hDek1;Dek2;Dek3iT

Ek¼Ek1;Ek2;Ek3T

ð3Þ The [6·6] arrayCekcontains the elastic, the [3·6] array ek the piezoelectric and finally the [3·3] array ek the dielectric coefficients of thek-th layer. For convenience the stresses and strains are separated into in-plane and transverse components, denoted respectively with p andnas follows:

rkp¼rk11;rk22;rk12T

rkn¼rk13;rk23;rk33T

ð4Þ kp¼k11; k22; k12T

kn¼k13; k23; k33T

ð5Þ The array formulation Eq.(1)of the constitutive equa- tions can then be rewritten as

rkp¼CekppkpþCekpnknekTpEk rkn¼CekTpnkpþCeknnkneknTEk Dek¼ekpkpþeknknþekEk

ð6Þ

The arrays of the elastic material properties for mono- clinic material systems explicitly read

e Ckpp¼

e

Ck11 Cek12 Cek16 Cek12 Cek22 Cek26 e

Ck16 Cek26 Cek66 2

66 64

3 77 75 Cekpn¼

0 0 Cek13 0 0 Cek23 0 0 Cek36 2

66 64

3 77 75

e Cknn¼

e

Ck55 Cek45 0 e

Ck45 Cek44 0 0 0 Cek33 2

66 64

3 77 75

The arrays of the piezoelectric constants for monoclinic material are

ekp¼

0 0 0

0 0 0

ek31 ek32 ek36 2

64

3 75 ekn¼

ek14 ek15 0 ek24 ek25 0 0 0 ek33 2

64

3 75

k=1 k=2 k=NL

k=N -1L

k

x,y z ,kzk

W Wk

x ,yk k

z0 k

h z

h

Fig. 1. Geometry and notations of multilayered plate.

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The piezoelectric materials considered in this work are all transversally isotropic to the 3-direction. Thus the componentse14,e25ande36become zero. The permittiv- ity coefficients are combined to the array

e11 e12 0 e21 e22 0 0 0 e33

2 64

3 75

Attention has been herein restricted to the case e12=e21= 0 which corresponds to hexagonal crystal sys- tems, see[21].

2.2. Geometric relations

The strainskpandknare related to the displacements uk¼ ½ukx;uky;ukzT according to the linear geometrical relations

kp¼Dpuk kn¼Dnuk ð7Þ

The arraysDpandDncontain the differential opera- tors. The definition of the electric field strength Ek according to the Maxwell equations is given by

Ek¼DeUk ð8Þ

The explicit form of the introduced linear differential operators arrays read

Dp¼

ox 0 0 0 oy 0 oy ox 0 2

64

3 75 Dn¼

oz 0 ox

0 oz oy

0 0 oz

2 64

3 75

De¼

ox 0 0

0 oy 0

0 0 oz

2 64

3 75

whereinoadenotes partial derivation with respect to the a-coordinate.

3. Unified formulation

The key point of the unified formulation is the usage of generalized assumptions for the displacementsuand, in the piezoelectric case, also for the potentialU. In the following section, these assumptions are defined and the possible theories are listed.

3.1. Assumptions for displacement

The generalized assumptions for the displacement u= [ux,uy,uz]Twith the given number of expansion N are

u¼FtutþFrurþFbub¼Fsus

wheres¼t;b;r and r¼1;2;. . .;N ð9Þ The displacement is separated into a set of thickness functions Fs and the correspondent displacement vari-

ablesus. This formulation incorporates a wide variety of different theories with the same basic arrays for the governing equations (fundamental nuclei). The two basic concepts are layer-wise (LW) and equivalent single layer (ESL) modeling. In the first case different displace- ment variables are assumed for every layer, while in the second case only one set of displacement variables for the whole plate is defined.Fig. 2shows from a qualita- tive point of view the differences between LW and ESL descriptions with linear (N= 1) or higher-order expan- sions for a three layer problem For convenience the two modelings (LW and ESL) have been in Fig. 2de- noted by LDN ed EDN, according to the acronyms that will be clarified below.

3.1.1. ESL model with Taylor expansion

The equivalent single layer description requires an assumption of the displacement for the whole plate.

Using Taylor expansions it can generally be written as u¼zrur r¼0;1;2;. . .;N ð10Þ Expressed in the form of the generalized assumption(9), the subscript b denotes values related to the reference surfaceX and the subscript t refers to the terms with the highest order (N). The thickness functions thus are the following Taylor expansions

Fb¼1; Ft¼zN; Fr¼zr; r¼1;2;. . .;N1 ð11Þ In this work we consider linear (N= 1) up to fourth- order (N= 4) models. These models are denoted as EDN where N indicates the order of expansion. For in- stance, for ED1 the explicit displacement assumptions read

Fig. 2. Displacement assumptions of ESL and LW model.

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ux¼uxbþzuxt

uy¼uybþzuyt

uz¼uzbþzuzt

The variable uib represents the displacement of the reference surface X and the variable uit is interpreted as the rotation angle/i. In this respect, the First order Shear Deformation Theory (FSDT) can be found by set- tinguzt=/z= 0. Furthermore, results for the Classical Laminate Theory (CLT) with the conditionuxt=/x= uz,x and uyt=/y=uz,y can be obtained by using the shear correction factorv=1within a typical pen- alty technique.

3.1.2. Inclusion of ZZ-functions in ESLM

A refinement of the ESL formulation can be reached by adding a function to the displacement assumption, that imposes the zig-zag form (ZZ) of the displacement distribution. According to Murakami[20], who first pre- sented this idea, the displacement with imposed ZZ form reads

u¼u0þ ð1ÞkfkuZþzrur; r¼1;2;. . .;N ð12Þ The zig-zag term, denoted with the subscriptZchanges sign for each layerk. In unified form this relation can be represented, if the subscripttis referred to the ZZ func- tion (ut=uZ). In this case, the thickness functions are defined as

Fb¼1; Ft¼ ð1Þkfk; Fr¼zr; r¼1;2;. . .;N ð13Þ These ESLM with ZZ functions are considered with lin- ear, parabolic and cubic expansion, referred to as EDZ1, EDZ2 and EDZ3. For example the explicit displacement assumptions of EDZ1 take the form

ux¼ux0þz/xþ ð1ÞkfkuxZ

uy¼uy0þz/yþ ð1ÞkfkuyZ

uz¼uz0þz/zþ ð1ÞkfkuzZ

The construction of EDZ1 is depicted inFig. 3.

3.1.3. LW model with Legendre expansions

In case of layer-wise modeling, the displacement vari- ables are assumed independently for each layerk. Thus a statement in the unified form

uk¼Ftukt þFbukbþFrukr ¼Fsuks

wheres¼t;b;r; r¼1;2;. . .;N and

k¼1;2;. . .;NL ð14Þ

exists for every layerk. For LW models it is much more convenient to use combinations of Legendre polynomi- als as thickness functions. As they offer the possibility to defineutandubas top and bottom values of the dis- placements of the layer, the interlaminar continuity can easily be implemented in the assembly of the laminate.

The thickness functions Fsfor the LW case are defined as

Fb¼ ðP0þP1Þ=2; Ft¼ ðP0P1Þ=2;

Fr ¼PrPr2 withr¼1;2;. . .;N ð15Þ in whichPj=Pj(fk) is the Legendre polynomial ofjth or- der defined over the layer thicknessfkwith16fk61.

The LW models range as well from linear up to fourth- order (LD1, LD2, LD3 and LD4). The employed poly- nomials are in the following explicitly given:

P0¼1; P1¼fk; P2¼ ð3f2k1Þ=2;

P3¼ ð5f3k3fkÞ=2; P4¼ ð35f4k30f2kþ3Þ=8

As the top and bottom values of the displacements have been chosen as unknown variables, the interlaminar compatibility can be imposed by

ukt ¼uðkþ1Þb ; k¼1;. . .;NL1 ð16Þ

3.2. Assumptions for potential

The modeling of the potentialUwill be restricted to layer-wise formulation. As hybrid problems containing piezoelectric and pure elastic layers are considered, the differences of the electric properties of each layer can be very significant. ESL assumptions thus do not seem appropriate to cover these high gradients. In contrast to the displacementu, the potentialUis a scalar. To ob- tain typical 3·3 matrices for the fundamental nuclei, it is useful for computational reasons that will be clarified later, to introduce the potential as a vectorUk= [Uk,Uk, Uk]T. Each component has the same value, the scalar potential U. With this assumption the layer-wise assumptions for the potential can be written in accor- dance to Eq.(14)as

Uk¼FtUkt þFbUkbþFrUkr ¼FsUks wheres¼t;b;r; r¼1;2;. . .;N

and k¼1;2;. . .;NL ð17Þ

The same thickness functions are used as in the layer- wise displacement case, stated in Eq.(15). The interlam- inar compatibility conditions of the potential can thus be imposed by

EDZ1:

+

z z z

ED1 ZZ EDZ1

=

Fig. 3. Displacement assumptions of EDZ1.

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Ukt ¼Uðkþ1Þb ; k¼1;. . .;NL1 ð18Þ For convenience the expansionNof the potential is as- sumed to be the same as the expansion of the displace- ment assumption, no matter if the latter is chosen to be ESL or LW.

4. Governing equations

The unified formulation of the differential equations and boundary conditions governing the coupled mechanical–electrical behavior of hybrid piezoelectric mechanic plates are derived in this section, using the principle of virtual displacements (PVD). The derivation is performed at layer-level, followed by the assembly of the arrays for the laminate. In the third part a Navier- type closed-form solution is presented.

4.1. Governing equations for a single layer

The PVD for a piezoelectric layerkin array formula- tion can be stated as

Z

Xk

Z

Ak

dkpTrkpþdknTrkndETDe

n o

dXkdz

¼ Z

Xk

Z

Ak

qkdukT€ukdXkdzþdWe ð19Þ

The external virtual work is represented as dWe. The geometrical relations (7) and (8) and the constitutive equations (6) are introduced and displacement and potential are expressed with the assumptions (9) and (17).

Z

Xk

Z

Ak

Dpduks T

Fs CekppDpþCekpnDn

FsuksekpTDeFsUks

h i

n

þ ðDnduksÞTFs CeknpDpþCeknnDn

FsukseknTDeFsUks

h i

ðDedUksÞTFs ekpDpþeknDn

FsuksþekDeFsUks

h io

dXkdz

¼ Z

Xk

Z

Ak

duksTFsqkFs€uksdXkdzþdWe ð20Þ

DnandDecontain the differential operators in both the in-plane and the transverse direction. In order to perform integration-by-parts in the domain Xk, these operators are conveniently splitted into their in-plane (subscriptX) and transverse (subscriptz) components:

Dn¼DnXþDnz ð21Þ

De¼DeXþDez ð22Þ

The integration-by-parts in Xk can be carried out according to the following array scheme:

Z

Xk

ðDndukÞTukdXk¼ Z

Xk

dukTDTnukdXk

þ Z

Ck

dukTInukdCk

wheren¼p;nX;eX ð23Þ

with

Ip¼

1 0 0

0 1 0

1 1 0

2 4

3 5; InX¼

0 0 1

0 0 1

0 0 0

2 4

3 5

and

IeX¼

1 0 0

0 1 0

0 0 0

2 4

3

5 ð24Þ

The integration yields following expression Z

Xk

Z

Ak

duksThDTpFsðCekppDpþCekpnðDnXþDnzÞÞFsuks n

ekpTðDeXþDezÞFsUks

þ ðDTnzDTnXÞFsðCeknpDpþCeknnðDnXþDnzÞÞFsuks eknTðDeXþDezÞFsUksi

dUksThðDTezDTeXÞFsðekpDpþeknðDnXþDnzÞÞFsuks þekðDeXþDezÞFsUksio

dXkdz þ

Z

Ck

Z

Ak

duksThITpFsðCekppDpþCekpnðDnXþDnzÞÞFsuks n

ekpTðDeXþDezÞFsUks

þITnXFsðCeknpDpþCeknnðDnXþDnzÞÞFsuks eknTðDeXþDezÞFsUksi

dUksThITeXFsððekpDpþeknðDnXþDnzÞÞFsuks þekðDeXþDezÞFsUksÞio

dCkdz

¼ Z

Xk

Z

Ak

duksTFsqkFs€uksdXkdzþdWe ð25Þ

From this equation, the coupled electro-mechanical sys- tem of governing equations can be obtained. The varia- tions of displacement duk and potential dUk are independent and can be therefore formulated in two separate equations

duks: KkssuuuksþKkssueUks¼ Mkss€uksþpkms ð26Þ

dUks: KksseuuksþKksseeUks ¼pkes ð27Þ with the boundary conditions

uks¼uks or PkssuuuksþPkssueUks¼PkssuuuksþPkssueUks Uks¼Uks or PksseuuksþPksseeUks ¼PksseuuksþPksseeUks

ð28Þ

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The unified formulation introduces the following fundamental nuclei

Kkssuu¼ Z

Ak

DTpCkppFsFsDpþCkpnFsFsDnXþCkpnFsDnzFs n

DTnX CknpFsFsDpþCknnFsFsDnXþCknnFsDnzFs

þDTnzFs CknpFsDpþCknnFsDnXþCknnDnzFs

o

dz ð29Þ Kkssue ¼

Z

Ak

DTp ekpTFsDezFsþekpTFsFsDeX

n

þDTnX eknTFsDezFsþeknTFsFsDeX

DTnzFs eknTDezFsþeknTFsDeX

o

dz ð30Þ

Kksseu ¼ Z

Ak

DTeX ekpFsFsDpþeknFsDnzFsþeknFsFsDnX

n

DTezFs ekpFsDpþeknDnzFsþeknFsDnX

o

dz ð31Þ

Kkssee ¼ Z

Ak

DTeXekFsDezFsþekFsFsDeX

DTezFs ekDezFsþekFsDeX

dz ð32Þ

The matrices for the boundary terms read Pkssuu¼

Z

Ak

IpðCkppFsFsDpþCkpnFsFsDnXþCkpnFsDnzFsÞ n

þInXðCknpFsFsDpþCknnFsFsDnXþCknnFsDnzFsÞo dz ð33Þ Pkssue ¼

Z

Ak

IpekpTFsDezFsþekpTFsFsDeX n

InX eknTFsDezFsþeknTFsFsDeX

o

dz ð34Þ

Pksseu ¼ Z

Ak

ITeX ekpFsFsDpþeknFsDnzFsþeknFsFsDnX

n o

dz ð35Þ Pkssee ¼

Z

Ak

ITeX ekFsDezFsþekFsFsDeX

dz ð36Þ

and the mass matrix can be calculated as Mkss¼

Z

Ak

fqkFsFsIgdz ð37Þ

The explicit form of the fundamental nuclei for mono- clinic material properties is stated inAppendix A.1.

4.2. Assembly of laminate arrays

In the previous section the fundamental matrices for piezoelectric layers have been derived. They are build

according to the scheme displayed inFig. 4, where the indices sandsgive the positions t,r andbas defined for the generalized assumptions. In case of displace- ments, each subparttt,tretc. consist of the three compo- nentsux,uyanduz. The scalar electric potential has been artificially expanded to a [3·1] vector, thus leading to a [3·3] fundamental nucleus analogous to the displace- ments. During assembly, the terms related to the poten- tial can be contracted back to their scalar nature. While Kkssuu has the dimension [3·3], Kkssue and Kksseu become [3·1] and [1·3] arrays, respectively, and Kkssee is reduced to a [1·1] matrix.

4.2.1. LW assembly

If the displacement is assumed layer-wise, the layer arraysKkssij with (ij) = (uu), (ue), (eu) and (ee), are com- bined to the laminate matrices Kssij according to the scheme in Fig. 5. The interface conditions for the dis- placements and the potential Eqs. (16) and (18) are imposed by superposing the related subparts as demon- strated in the figure.

4.2.2. ESL assembly

In case of ESL modeling only one set of displace- mentsuexists. Thus the layer arraysKkssuu are all super- posed on this displacement. The potential is still modeled LW, so that forKkssue andKksseu only the displace- ments and the top and bottom potentials at each layer

K

k tt

t= t

r b

s = t

r b

K

k tt

K

k tr

K

k tb

K

k rb

K

k rr

K

k rt

K

k bb

K

k br

K

k bt

K

k st

:

Fig. 4. Basic scheme of layer arraysKkssij (i,j=u,e).

k=NL

k=N -1L

t r b r b

t r b r b

b =t

k k -1

Fig. 5. LW assembly of laminate arraysKssij (i,j=u,e).

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interface are superposed. At lastKkssee is assembled as in the pure LW case. This assembly procedure is depicted inFig. 6.

4.3. Closed-form solution

Closed-form solution to the problem given by Eqs.

(26)–(28)can be found, if some restrictions are applied.

The considered plate must be simply-supported which implies the following boundary conditions on the displacements:

ukxðx;y;zÞ ¼0 aty¼0;b ukyðx;y;zÞ ¼0 atx¼0;a

ukzðx;y;zÞ ¼0 atx¼0;a y¼0;b

ð38Þ

In addition each layer is orthotropic in the laminate refe- rence system, i.e. the components Ce16;Ce26;Ce36andCe45

vanish. In this case, the following Navier-type assump- tions can be made

ukxs;pkmxs

¼^ukxs;^pkmxs

cosðaxÞsinðbyÞeixmnt ukys;pkmys

¼

^ ukys;^pkmys

sinðaxÞcosðbyÞeixmnt ukzs;pkmzs

¼^ukzs;^pkmzs

sinðaxÞsinðbyÞeixmnt

ð39Þ

and Uks;pkes

¼U^ks;^pkes

sinðaxÞsinðbyÞeixmnt ð40Þ with the abbreviations

a¼mp

a and b¼np

b ð41Þ

The factorsmandnare the number of waves inx- and y-direction whileaandbare the length and width of the

plate. Introducing these assumptions in the system of governing equations and carrying out the differentia- tions, the fundamental nuclei for closed-form solution K^kssuu; K^kssue; K^ksseu andK^kssee can be found. It turns out that K^ksseu ¼K^kss

T

ue . Explicit forms of these arrays can be taken fromAppendix A.2. The assembly of the laminate ar- rays works in the same manner as described above. Thus the linear system for closed-form solution states K^uu^uþK^ueU^ ¼x2mnM^uþ^pm ð42Þ

K^Tue^uþK^eeU^¼^pe ð43Þ The dynamic free vibration frequencies of this problem can be found by solving the eigenvalue problem

K^uu K^ue K^ee

1

K^Tue

x2mnM

¼0 ð44Þ

Static responses to mechanical and electrical loadings^pm and^pecan be calculated if the loadings are in the form of Eqs.(39) and (40). Boundary conditions on the potential U^ can be imposed on the top and bottom surface and at the layer interfaces; in all cases,U^ is continuous alongz.

5. Numerical results

Three different problems have been addressed. The first is a dynamic analysis and the second and third con- sider the static response of a plate on two different load- ing and boundary condition configurations. For all three problems 3D exact solutions are available, which are used for the verification of the presented model.

5.1. Definition of problems 5.1.1. Problem I

In this testcase a square plate with the length and widthaand the thicknesshis considered, seeFig. 7. It consists of five layers which are assumed to be perfectly bonded to each other. The top and bottom layers are made of piezoelectric materials with the thickness

kts

Kee

kts

t r b

t r b

Σ

Ku u

k =1

NL

δ

u

u

kts

Ku e

k=NL N -1L N -2L

t r b

Φ

K

u ut s

K

u et s

k = NL N -1L N -2L

δΦ

K

eut s

t r b

kts

Keu

k = NL

N -1L N -2L

k=NL N -1L N -2L

K

eut s

Fig. 6. ESL assembly of laminate arraysKssij (i,j=u,e).

x y

z

h

a

Φt= 0

Φb= 0

a

Fig. 7. Problem I, geometry and boundary conditions.

(10)

hkp¼0:1heach, the three structural layers of equal thick- ness have the configuration [0/90/0]. The materials are PZT-4 for the piezoelectric and Gr/Ep for the structural layers. The corresponding material properties can be taken fromTable 1. The top and bottom surfaces are assumed to be traction-free and electrically grounded, which imposes the electric potential to be zero (Ut= Ub= 0). The free vibration frequencies for one wave in each direction (m=n= 1) are calculated in means of the frequency parameter c=x/100. Four thickness ratiosa/h= 2, 4, 10 and 50 are considered. A 3D exact solution is available from Heyliger and Saravanos[22]

for the casesa/h= 4 and 50. Further numerical results for these cases can be taken from Benjeddou [15]and Touratier and Ossadzow-David[13].

5.1.2. Problem II

The configuration for Problem II and III consists of a simply supported square plate with the side lengthaand the thicknessh. It is build of four layers, two layers of fiber reinforced material as a structural core on top and bottom of which the two piezoelectric layers are bonded. The core layers have the thickness hc= 0.4h each and the fiber direction is in [0/90] configuration.

The thickness of each piezoelectric layer is hp= 0.1h.

The materials are PZT-4 and Gr/Ep with the same prop- erties as used for Problem I (Table 1). The analytic solu- tion is calculated for one wave in each direction (m=n= 1). Four different thickness ratios a/h= 2, 4, 10 and 100 are considered.

For Problem II, a mechanical loading of^pz¼1 is ap- plied on the top surface of the plate. The top and bottom surfaces are electrically grounded and thus have the po- tential Ut=Ub= 0. The configuration of Problem II is shown inFig. 8. 3D exact solution to Problems II and III fora/h= 4 were presented by Heyliger[23].

5.1.3. Problem III

The plate of Problem III is the same as for Problem II and again a closed-form solution is determined for m=n= 1. Only the boundary conditions and loading configurations are changed as shown in Fig. 9. The Table 1

Elastic, piezoelectric and dielectric properties of used materials

Property PZT-4 Gr/EP

E1[GPa] 81.3 132.38

E2[GPa] 81.3 10.756

E3[GPa] 64.5 10.756

m12[–] 0.329 0.24

m13[–] 0.432 0.24

m23[–] 0.432 0.49

G44[GPa] 25.6 3.606

G55[GPa] 25.6 5.6537

G66[GPa] 30.6 5.6537

e15[C/m2] 12.72 0

e24[C/m2] 12.72 0

e31[C/m2] 5.20 0

e32[C/m2] 5.20 0

e33[C/m2] 15.08 0

~11=0[–] 1475 3.5

~22=0[–] 1475 3.0

~33=0[–] 1300 3.0

x y

z

h

a

p x yz( , )=p^zsin( )sin( )

Φt= 0 Φb= 0 a

πx πy

a a

Fig. 8. Problem II, geometry, boundary conditions and loading configuration.

x y

z

h

a

^

Φb= 0

Φt( , )=x y Φtsin( )sin( )

a

πx πy

a a

Fig. 9. Problem III, geometry, boundary conditions and loading configuration.

Table 2

Verification—Problem I, frequency parametersc=x/100 of exact solution and LD4

a/h Mode 1 Mode 2 Mode 3 Mode 4 Mode 5 Mode 6

4 Exact 57074.5 191301 250769 274941 362492 381036

LD4 57074.0 191301 250768 274940 362489 381036

50 Exact 618.118 15681.6 21492.8 209704 210522 378104

LD4 618.104 15681.6 21492.6 209704 210522 378104

(11)

top and bottom surfaces are now traction free and a po- tential of U^t¼1 is imposed on the top surface of the plate. The bottom surface remains grounded and thus has the potentialUb= 0.

5.2. Verification

To verify the presented model and to assess its accu- racy a comparison with results from the 3D exact Table 3

Verification—Problem II, thickness distribution ofU,rxxandDeof exact solution and LD4

z U·103 rxx De

Exact LD4 Exact LD4 Exact LD4

0.500 0.0000 0.000 6.5643 6.5642 160.58 160.59

0.475 0.0189 0.189 5.8201 5.8200 149.35 149.35

0.450 0.0358 0.352 5.0855 5.0855 117.23 117.23

0.425 0.0488 0.488 4.3595 4.3595 66.568 66.567

0.400 0.0598 0.598 3.6408 3.6408 0.3382 0.3348

0.400 0.0598 0.598 2.8855 2.8858 0.3382 0.3384

0.300 0.0589 0.590 1.4499 1.4496 0.1276 0.1277

0.200 0.0589 0.589 0.2879 0.2880 0.0813 0.0813

0.100 0.0596 0.593 0.7817 0.7814 0.2913 0.2914

0.000 0.0611 0.611 1.9266 1.9266 0.5052 0.5053

0.000 0.0611 0.611 0.0991 0.0991 0.5052 0.5053

0.100 0.0634 0.634 0.0149 0.0150 0.7259 0.7260 0.200 0.0665 0.666 0.1280 0.1281 0.9563 0.9565 0.300 0.0706 0.706 0.2426 0.2427 1.1995 1.1997 0.400 0.0756 0.756 0.3616 0.3617 1.4587 1.4589 0.400 0.0756 0.756 4.2348 4.2348 1.4587 1.4559 0.425 0.0602 0.602 4.8806 4.8806 58.352 58.351 0.450 0.0425 0.425 5.5337 5.5337 103.66 103.67 0.475 0.0224 0.225 6.1951 6.1950 132.40 132.41 0.500 0.0000 0.000 6.8658 6.8658 142.46 142.46

Table 4

Verification—Problem III, thickness distribution ofux,rxxandUof exact solution and LD4

z ux·1012 rxx U

Exact LD4 Exact LD4 Exact LD4

0.500 32.764 32.765 111.81 111.80 1.0000 1.0000

0.475 23.349 23.350 63.736 63.738 0.9971 0.9972

0.450 13.973 13.974 15.833 15.839 0.9950 0.9951

0.425 4.6174 4.6179 32.001 31.992 0.9936 0.9936

0.400 4.7356 4.7353 79.865 79.852 0.9929 0.9929

0.400 4.7356 4.7353 51.681 51.680 0.9929 0.9929

0.300 2.9808 2.9802 33.135 33.127 0.8415 0.8416

0.200 1.7346 1.7346 19.840 19.840 0.7014 0.7015

0.100 0.8008 0.8014 9.7737 9.7780 0.5707 0.5708

0.000 0.0295 0.0297 1.3905 1.3953 0.4476 0.4477

0.000 0.0295 0.0297 1.3089 1.3099 0.4476 0.4477

0.100 0.4404 0.4401 0.5782 0.5778 0.3305 0.3306 0.200 0.8815 0.8812 0.1348 0.1343 0.2179 0.2179 0.300 1.3206 1.3202 0.8463 0.8465 0.1081 0.1082 0.400 1.7839 1.7835 1.5723 1.5708 0.0010 0.0010 0.400 1.7839 1.7835 14.529 14.524 0.0010 0.0010 0.425 2.0470 2.0465 178.01 17.793 0.0009 0.0010 0.450 2.3140 2.3134 210.98 21.090 0.0008 0.0008 0.475 2.5856 2.5850 244.28 24.418 0.0004 0.0004 0.500 2.8625 2.8618 277.95 27.784 0.0000 0.0000

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